Spectral coherence of white LEDs

Atri Halder; Jari Turunen

doi:10.1364/PRJ.461314

Photonics Research, Volume. 10, Issue 11, 2460(2022)

Spectral coherence of white LEDs

Atri Halder^* and Jari Turunen

Author Affiliations

Institute of Photonics, University of Eastern Finland, FI-80101 Joensuu, Finland

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Equations(53)

W (\bar{ρ}, Δ ρ, ω) = S (\bar{ρ}, ω) μ (Δ ρ, ω),

(1)

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T (κ_{1}, κ_{2}, ω) = \frac{1}{{(2 π)}^{4}} \iint_{- \infty} W (ρ_{1}, ρ_{2}, ω) \times \exp [i (κ_{1} \cdot ρ_{1} - κ_{2} \cdot ρ_{2})] d^{2} ρ_{1} d^{2} ρ_{2},

(2)

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T (\bar{κ}, Δ κ, ω) = \frac{1}{{(2 π)}^{4}} \iint_{- \infty} W (\bar{ρ}, Δ ρ, ω) \times \exp [- i (Δ κ \cdot \bar{ρ} + \bar{κ} \cdot Δ ρ)] d^{2} \bar{ρ} d^{2} Δ ρ

(3)

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T (\bar{κ}, Δ κ, ω) = \tilde{S} (\bar{κ}, ω) \tilde{μ} (Δ κ, ω),

(4)

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\tilde{S} (Δ κ, ω) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} S (\bar{ρ}, ω) \exp (- i Δ κ \cdot \bar{ρ}) d^{2} \bar{ρ}

(5)

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\tilde{μ} (\bar{κ}, ω) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} μ (Δ ρ, ω) \exp (- i \bar{κ} \cdot Δ ρ) d^{2} Δ ρ

(6)

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W^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = {(\frac{2 π ω}{rc})}^{2} s_{z 1} s_{z 2} T (\frac{ω}{c} s_{⊥ 1}, \frac{ω}{c} s_{⊥ 2}, ω),

(7)

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W^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = {(\frac{2 π ω}{rc})}^{2} s_{z 1} s_{z 2} \tilde{S} (\frac{ω}{c} Δ s_{⊥}, ω) \tilde{μ} (\frac{ω}{c} {\bar{s}}_{⊥}, ω) .

(8)

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S^{(\infty)} (r \hat{s}, ω) = {(\frac{2 π ω}{r c})}^{2} s_{z}^{2} \tilde{S} (0, ω) \tilde{μ} (\frac{ω}{c} Δ s_{⊥}, ω)

(9)

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μ^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = \frac{\tilde{S} [(ω / c) Δ s_{⊥}, ω]}{\tilde{S} (0, ω)},

(10)

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\tilde{S} (0, ω) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} S (ρ, ω) d^{2} ρ

(11)

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J (\hat{s}, ω) = \lim_{r \to \infty} r S^{(\infty)} (r \hat{s}, ω) = {(2 π)}^{2} s_{z}^{2} {(ω / c)}^{2} \tilde{S} (0, ω) \tilde{μ} (\frac{ω}{c} s_{⊥}, ω)

(12)

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μ (Δ ρ, ω) = \int_{| κ | < ω / c} \tilde{μ} (κ, ω) \exp (i κ \cdot Δ ρ) d^{2} κ .

(13)

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μ (Δ ρ, ω) = \frac{1}{{(2 π)}^{2} \tilde{S} (0, ω)} \times \int_{LF} \frac{J (\hat{s}, ω)}{1 - s_{⊥}^{2}} \exp (i \frac{ω}{c} s_{⊥} \cdot Δ ρ) d^{2} s_{⊥},

(14)

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μ (Δ ρ, ω) = \frac{\int_{LF} {(1 - s_{⊥}^{2})}^{- 1} J (\hat{s}, ω) \exp [i (ω / c) s_{⊥} \cdot Δ ρ] d^{2} s_{⊥}}{\int_{LF} {(1 - s_{⊥}^{2})}^{- 1} J (\hat{s}, ω) d^{2} s_{⊥}} .

(15)

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μ (Δ ρ, ω) = \frac{\int_{0}^{1} s {(1 - s^{2})}^{- 1} J (s, ω) J_{0} [(ω / c) Δ ρ s] d s}{\int_{0}^{1} s {(1 - s^{2})}^{- 1} J (s, ω) d s},

(16)

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μ (Δ ρ, ω) = \frac{1}{C^{2}} \int_{- \infty}^{\infty} f^{*} (\bar{ρ} - ρ^{'} - Δ ρ / 2, ω) \times f (\bar{ρ} - ρ^{'} + Δ ρ / 2, ω) d^{2} ρ^{'},

(17)

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C^{2} = \int_{- \infty}^{\infty} {| f (ρ, ω) |}^{2} d^{2} ρ .

(18)

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W (ρ_{1}, ρ_{2}, ω) = f^{*} (ρ_{1}, ω) f (ρ_{2}, ω)

(19)

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\tilde{f} (κ, ω) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} f (ρ, ω) \exp (- i κ \cdot ρ) d^{2} ρ .

(20)

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J (\hat{s}, ω) = {(2 π)}^{2} s_{z}^{2} {(ω / c)}^{2} {| \tilde{f} (\frac{ω}{c} s_{⊥}, ω) |}^{2} .

(21)

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f (ρ, ω) = \frac{1}{2 π} \int_{LF} \frac{J^{1 / 2} (\hat{s}, ω)}{{(1 - s_{⊥}^{2})}^{1 / 2}} \exp (i \frac{ω}{c} s_{⊥} \cdot ρ) d^{2} s_{⊥},

(22)

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f (ρ, ω) = f (0, ω) \frac{\int_{0}^{1} s {(1 - s^{2})}^{- 1 / 2} J^{1 / 2} (s, ω) J_{0} [(ω / c) ρ s] d s}{\int_{0}^{1} s {(1 - s^{2})}^{- 1 / 2} J^{1 / 2} (s, ω) d s},

(23)

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J (\hat{s}, ω) = J (ω) s_{z}^{p} = J (ω) \cos^{p} θ,

(24)

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\int_{0}^{1} x {(1 - x^{2})}^{μ} J_{0} (a x) d x {= 2}^{μ} Γ (μ + 1) \frac{J_{μ + 1} (a)}{a^{μ + 1}}

(25)

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\int_{0}^{1} x {(1 - x^{2})}^{μ} d x = \frac{1}{2 (1 + μ)},

(26)

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μ (Δ ρ, ω {) = 2}^{p / 2} Γ (1 + \frac{p}{2}) \frac{J_{p / 2} [(ω / c) Δ ρ]}{{[(ω / c) Δ ρ]}^{p / 2}}

(27)

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f (ρ, ω) = f (0, ω {) 2}^{q / 4 + 1 / 2} Γ (\frac{3}{2} + \frac{q}{4}) \frac{J_{q / 4 + 1 / 2} [(ω / c) ρ]}{{[(ω / c) ρ]}^{q / 4 + 1 / 2}}

(28)

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μ^{'} (Δ ρ^{'}, ω) = \frac{\int_{0}^{{NA}^{'}} s^{'} {(1 - s^{' 2})}^{- 1} J^{'} (s^{'}, ω) J_{0} [(ω / c) Δ ρ^{'} s^{'}] d s^{'}}{\int_{0}^{{NA}^{'}} s^{'} {(1 - s^{' 2})}^{- 1} J^{'} (s^{'}, ω) d s^{'}}

(29)

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\frac{f^{'} (ρ^{'}, ω)}{f^{'} (0, ω)} = \frac{\int_{0}^{{NA}^{'}} s^{'} {(1 - s^{' 2})}^{- 1 / 2} J^{' 1 / 2} (s^{'}, ω) J_{0} [(ω / c) ρ^{'} s^{'}] d s^{'}}{\int_{0}^{{NA}^{'}} s^{'} {(1 - s^{' 2})}^{- 1 / 2} J^{' 1 / 2} (s^{'}, ω) d s^{'}}

(30)

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M = \frac{F^{'}}{F} = \frac{\sin θ}{\sin θ^{'}} = \frac{NA}{{NA}^{'}} = \frac{\sin Θ}{\sin Θ^{'}} = \frac{s}{s^{'}} = \frac{Δ ρ^{'}}{Δ ρ} = \frac{ρ^{'}}{ρ} .

(31)

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J (θ, ω) \cos θ = J^{'} (θ^{'}, ω) \cos θ^{'}

(32)

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J^{'} (s^{'}, ω) = J (s, ω) {[\frac{1 - s^{2}}{1 - {(s / M)}^{2}}]}^{1 / 2} = J (s, ω) P (s),

(33)

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μ^{'} (M Δ ρ, ω) = \frac{\int_{0}^{NA / M} s {[1 - {(s / M)}^{2}]}^{- 1} P (s) J (s, ω) J_{0} [(ω / c) Δ ρ s] d s}{\int_{0}^{NA / M} s {[1 - {(s / M)}^{2}]}^{- 1} P (s) J (s, ω) d s}

(34)

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\frac{f^{'} (M ρ, ω)}{f^{'} (0, ω)} = \frac{\int_{0}^{NA / M} s {[1 - {(s / M)}^{2}]}^{- 1 / 2} P (s) J^{1 / 2} (s, ω) J_{0} [(ω / c) ρ s] d s}{\int_{0}^{NA / M} s {[1 - {(s / M)}^{2}]}^{- 1 / 2} P (s) J^{1 / 2} (s, ω) d s},

(35)

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s^{(\infty)} (r \hat{s}, ω) = \frac{S^{(\infty)} (r \hat{s}, ω)}{\int_{0}^{\infty} S^{(\infty)} (r \hat{s}, ω) d ω} = \frac{J (θ, ω)}{\int_{0}^{\infty} J (θ, ω) d ω} .

(36)

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s^{(\infty)} (r \hat{s}, ω) = \frac{J (ω)}{\int_{0}^{\infty} J (ω) d ω},

(37)

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μ (Δ ρ, ω) = H (\frac{ω}{c} Δ ρ) = \frac{\int_{0}^{1} s {(1 - s^{2})}^{- 1} J (s) J_{0} [(ω / c) Δ ρ s] d s}{\int_{0}^{1} s {(1 - s^{2})}^{- 1} J (s) d s} .

(38)

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s (ω) = \frac{\tilde{S} (0, ω)}{\int_{0}^{\infty} \tilde{S} (0, ω) d ω},

(39)

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\frac{f (ρ, ω)}{f (0, ω)} = F (\frac{ω}{c} ρ) = \frac{\int_{0}^{1} s {(1 - s^{2})}^{- 1 / 2} J^{1 / 2} (s) J_{0} [(ω / c) ρ s] d s}{\int_{0}^{1} s {(1 - s^{2})}^{- 1 / 2} J^{1 / 2} (s) d s},

(40)

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μ^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = \frac{\sin [(ω / c) L Δ s_{x} / 2]}{(ω / c) L Δ s_{x} / 2} \frac{\sin [(ω / c) L Δ s_{y} / 2]}{(ω / c) L Δ s_{y} / 2},

(41)

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J (s, ω) = S_{B} (ω) H (s, ω_{B}) + S_{Y} (ω) H (s, ω_{Y}),

(42)

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H (s, ω_{B}) = H (0, ω_{B}) \cos^{m} θ = C_{B} {(1 - s^{2})}^{m / 2},

(43)

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H (s, ω_{Y}) = H (0, ω_{Y}) \cos^{n} θ = C_{Y} {(1 - s^{2})}^{n / 2},

(44)

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μ (Δ ρ, ω) = \frac{m^{- 1} C_{B} S_{B} (ω) L_{B} (Δ ρ, ω) + n^{- 1} C_{Y} S_{Y} (ω) L_{Y} (Δ ρ, ω)}{m^{- 1} C_{B} S_{B} (ω) + n^{- 1} C_{Y} S_{Y} (ω)},

(45)

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L_{B} (Δ ρ, ω {) = 2}^{m / 2} Γ (1 + \frac{m}{2}) \frac{J_{m / 2} [(ω / c) Δ ρ]}{{[(ω / c) Δ ρ]}^{m / 2}},

(46)

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L_{Y} (Δ ρ, ω {) = 2}^{n / 2} Γ (1 + \frac{n}{2}) \frac{J_{n / 2} [(ω / c) Δ ρ]}{{[(ω / c) Δ ρ]}^{n / 2}} .

(47)

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\frac{f (ρ, ω)}{f (0, ω)} = \frac{{(1 + m / 2)}^{- 1} C_{B} S_{B} (ω) K_{B} (ρ, ω) + {(1 + n / 2)}^{- 1} C_{Y} S_{Y} (ω) K_{Y} (ρ, ω)}{{(1 + m / 2)}^{- 1} C_{B} S_{B} (ω) + {(1 + n / 2)}^{- 1} C_{Y} S_{Y} (ω)},

(48)

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L_{B} (ρ, ω {) = 2}^{m / 4 + 1 / 2} Γ (\frac{3}{2} + \frac{m}{4}) \frac{J_{m / 4 + 1 / 2} [(ω / c) ρ]}{{[(ω / c) ρ]}^{m / 4 + 1 / 2}},

(49)

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L_{Y} (ρ, ω {) = 2}^{n / 4 + 1 / 2} Γ (\frac{3}{2} + \frac{n}{4}) \frac{J_{n / 4 + 1 / 2} [(ω / c) ρ]}{{[(ω / c) ρ]}^{n / 4 + 1 / 2}} .

(50)

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S_{j} (ω) = \exp [- \frac{2}{Ω_{j}^{2}} {(ω - ω_{j})}^{2}]

(51)

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S_{j} (ω) = \exp [- \frac{2}{Ω_{j}^{2}} {(\frac{ω - ω_{0}}{ω_{s}} - ω_{j})}^{2}],

(52)

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ω_{0} = \frac{\int_{0}^{\infty} ω J (0, ω) d ω}{\int_{0}^{\infty} J (0, ω) d ω}

(53)

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Atri Halder, Jari Turunen, "Spectral coherence of white LEDs," Photonics Res. 10, 2460 (2022)

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Paper Information

Category: Physical Optics

Received: Apr. 19, 2022

Accepted: Aug. 7, 2022

Published Online: Oct. 8, 2022

The Author Email: Atri Halder (atri.halder@uef.fi)

DOI:10.1364/PRJ.461314

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology